Geometric proof shows the random proposer mechanism achieves a 4-approximation in bilateral exchange, refining to 3.15.
We provide a geometric proof that the random proposer mechanism is a $4$-approximation to the first-best gains from trade in bilateral exchange. We then refine this geometric analysis to recover the state-of-the-art approximation ratio of $3.15$.
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Hartline et al. (2025) studied this question.
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